The designation "space-time", suggesting that there are 3 space and one time dimensions, is somewhat misleading; we should rather speak of 4 space dimensions, which together allow time (or movement, energy) to exist! Time is the measure of movement, telling us how fast a determined movement, e.g. the rotation of the earth, is, compared to another movement, e.g. the physicist's clock.
That our Universe is 4-dimensional is obvious from Albert Einstein's mass-energy equivalence $ E = mc^2 $ and the relativistic energy invariant $ E^2/c^2 - p⃗^2 = m_0^2 c^2 $. Measuring distance in light-seconds instead of meters, the speed-of-light c becomes =1, and we get the simple formula:
$ E^2 = m^2 = m_0^2 + p⃗^2 = m_0^2 + p_1^2 + p_2^2 + p_3^2 $
From this formula it is obvious that the rest mass $m_0$ is the fourth component of the momentum (or movement) vector, and that the energy or mass is the total length (absolute value) of the momentum vector.
Mass is energy, mass is movement (of something more basic, certainly)!
Now, how is movement (or time) possible after all in the Universe? Despite of some Greek philosopher's denial of the possibility of movement, Leonhard Euler, a 18th century Swiss mathematician working in St Peterburg (Russia) found a surprising identity, stating that a sum of four squares can always be written as the product of two sums of each four squares. (This holds also for sums of two squares, and for sums of eight squares, but for nothing more - A. Hurwitz, 1895)
Applying this formula to our momentum vector, we can write:
$ (m_0^2 + p_1^2 + p_2^2 + p_3^2) = (r_0^2 + r_1^2 + r_2^2 + r_3^2)(M_0^2 + P_1^2 + P_2^2 + P_3^2) $
wherein the components are (proof by algebraic evaluation):
$ m_0 = (r_0M_0 - r_1P_1 - r_2P_2 - r_3P_3) $
$ p_1 = (r_0P_1 + r_1M_0 + r_2P_3 - r_3P_2) $
$ p_2 = (r_0P_2 - r_1P_3 + r_2M_0 + r_3P_1) $
$ p_3 = (r_0P_3 + r_1P_2 - r_2P_1 + r_3M_0) $
Now, let's suppose that the vector
$ P⃗ = (M_0, P_1, P_2, P_3) $ is a previous physical state, and
the vector $ R⃗ = (r_0, r_1, r_2, r_3) $ represents a physical process, transforming vector $ P⃗ $ into vector $ p⃗ = (m_0, p_1, p_2, p_3) $, and let's further assume that process $ R⃗ $ does not change the total energy of the system, i.e. $ (r_0^2 + r_1^2 + r_2^2 + r_3^2 ) = 1 $ , then the equation:
$ p⃗ = R⃗ * P⃗ $ , wherein * is the multiplication as defined above, or, given the bilinearity of Euler's 4-squares identity, also the equation:
$ p⃗ = p1⃗ + p2⃗ = R⃗ * (P1⃗ + P2⃗) = R⃗ * P ⃗ $
describe the movement or evolution of a system $ P⃗ $ or of
a system $ (P1⃗ + P2⃗) $ under the influence of a physical process $ R⃗ $, wherein the total energy is conserved.
The concept of time in physics is linked to such processes, implying and requireing all 4 dimensions of 4-space. Rest mass $ m_0$ itself is such a process!
